The power of linear programming for general-valued CSPs

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrised by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimisation problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors (arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213) to appear in SIAM Journal on Computing (SICOMP

The complexity of finite-valued CSPs

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, $\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as a sum of functions from $\Gamma$. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite size. We show that every constraint language $\Gamma$ either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or $\Gamma$ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$

C*-algebras associated with endomorphisms and polymorphisms of compact abelian groups

A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the $K$-theory of these algebras and use it to compute the $K$-groups in a number of interesting examples.Comment: 25 page

Genome-wide patterns of polymorphism in an inbred line of the African malaria mosquito Anopheles gambiae.

Anopheles gambiae is a major mosquito vector of malaria in Africa. Although increased use of insecticide-based vector control tools has decreased malaria transmission, elimination is likely to require novel genetic control strategies. It can be argued that the absence of an A. gambiae inbred line has slowed progress toward genetic vector control. In order to empower genetic studies and enable precise and reproducible experimentation, we set out to create an inbred line of this species. We found that amenability to inbreeding varied between populations of A. gambiae. After full-sib inbreeding for ten generations, we genotyped 112 individuals--56 saved prior to inbreeding and 56 collected after inbreeding--at a genome-wide panel of single nucleotide polymorphisms (SNPs). Although inbreeding dramatically reduced diversity across much of the genome, we discovered numerous, discrete genomic blocks that maintained high heterozygosity. For one large genomic region, we were able to definitively show that high diversity is due to the persistent polymorphism of a chromosomal inversion. Inbred lines in other eukaryotes often exhibit a qualitatively similar retention of polymorphism when typed at a small number of markers. Our whole-genome SNP data provide the first strong, empirical evidence supporting associative overdominance as the mechanism maintaining higher than expected diversity in inbred lines. Although creation of A. gambiae lines devoid of nearly all polymorphism may not be feasible, our results provide critical insights into how more fully isogenic lines can be created